There are several steps to unlocking this achievement. To begin with, you need to hit the three yellow targets in the middle/top of the table. Each time you hit one, the purple circle will spin.
Once you have hit all three targets the circle will raise up and show a ramp on the left with the letters S T A I R W A Y and a hole on the right of it. For this achievement you will want to be hitting the ball up the ramp on the left to spell out STAIRWAY.
Now that you have spelt STAIRWAY, aim for the CITADEL ramp and then the next ramp that is highlighted for you (I cannot quite remember which ramp it was sorry).
Finally, you will now need to hit the STAIRWAY ramp 3 times. You get around 10 seconds or so in between successful attempts. The best way I found was to catch the ball with the right flipper and aim for the ramp from there.
If you can hit the ramp 3 times in the time limit you will have completed the Stairway mode and your achievement will follow.
If your ball drains or you accidentally start a mission, your progress in spelling STAIRWAY will carry over to your next ball/next time you raise the circle, but if you fail to hit the ramp 3 times after activating Stairway mode, you will have to spell out STAIRWAY again.
One side note, when I unlocked this achievement, the mode was started by Whisper giving a random reward for sinking the ball in the Whisper hole 4 times.
Hope this helps
EDIT: this seems to be an alternative way to get this to unlock, with credit to Capta1n0bvious3
'Once I spelled Stairway. before I could proceed I needed to hit the Whisper hole. Note that this happened when the stairway was down, not up. Once I hit the Whisper hole, the stairway popped back up, I hit it in there 3 times and got the achievement.
BTW I never need to hit the Citadel ramp. Also Whisper can be seen in her hole.'